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Deriviation of WKB approximation

  1. Mar 10, 2007 #1
    Hey!

    In deriving the WKB approximation the wave function is written as

    [tex]
    \psi \left( x \right) = exp\left[ i S\left( x \right) \right ]
    [/tex]

    Now, in some of the deriviations I've seen, the function S(x) is expanded as a power series in [tex]\hbar[/tex] as

    [tex]
    S(x) = S_0(x) + \hbar S_1(x) + \frac{\hbar}{2} S_2(x) ...
    [/tex]

    I don't really understand this. It's something like [tex]S_0[/tex] being the classical result and, the next term being a first order quantum correction and so on. But why do you choose to expand in powers of [tex]\hbar[/tex]? Can somebody explain to me what this is all about?

    Thanks in advance
    René
     
  2. jcsd
  3. Mar 10, 2007 #2

    Tom Mattson

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    That particular form for [itex]S(x)[/itex] has the correspondence principle built right into it. If you take the limit as [itex]\hbar \rightarrow 0[/itex], you recover the classical result.
     
  4. May 21, 2008 #3
    (This thread appeared on Google and I have the exact same question) I am extremely confused at your statement. [itex]\hbar[/itex] is a constant, right? How on earth can one construct a power series of a function S(x) by expanding it as a function of a constant? What does that even mean?

    I have taken a few (more like 1.5 and some self study) classes in QM on the engineering side, but this is over my head. I've currently borrowed a few different QM textbooks and they all say the same opaque thing.
     
    Last edited: May 21, 2008
  5. May 21, 2008 #4
    You'll be seeing a lot more constants being treated like parameters in physics, so you'll have to get used to it.

    Lets parameterize all the possible universes by different values of [tex]\hbar[/tex], and solve quantum mechanics in each of them. Then you'd get a family of solutions parameterized by [tex]\hbar[/tex]. If you choose our universe, corresponding to our [tex]\hbar[/tex], then in principle you have the solution to QM in our universe, no?

    There's a caveat, of course. You have to assume that the solutions to problems behave smoothly with [tex]\hbar[/tex], which is a reasonable assumption, but only comes from experience.

    Anyway, if you stick around long enough you'll get to differentiate with respect to orbital angular momentum [tex]\ell[/tex] and all sorts of goodness (Feynman-Hellman theorem)
     
  6. May 21, 2008 #5
    I think I understand a little better now, but I'll try to explain what is bugging me still. After reading around, I've come to the conclusion a power series with respect to constants is not so far-fetched: For example, any decimal number can be expressed as a power series in 10, or any other number really.

    However, the textbook I am primarily using ( Bransden & Joachain Quantum Mechanics: Second Edition ) mentions the power series for S(x) "does not converge, but is an asymptotic series for the function S(x). As a result, the best approximation to S(x) is obtained by keeping a finite number of terms". I've been reading about asymptotic series, but their rationale/use isn't very clear to me still.
     
  7. May 21, 2008 #6
    I guess the closest I can come to what an asymptotic series means in words, it's "Adding more terms to the expansion won't make the relative error appreciably smaller". Is this accurate?
     
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